Proof:
The equivalent fractional VIE to the Cauchy problem (REF )-() in the space \(C_{1-\zeta \,; \,\Psi }(\Lambda ,\,\mbox{R})\)  is derived in [1]} and it is given by
\(y (t) =\mathcal {H}^{\Psi }_{\zeta }(t,a)\,y_{a}+\frac{1}{\Gamma ( \eta ) }\int _{a}^{t}\mathrm {Q}^{\eta \,;\,\psi }(t,\sigma )f(\sigma ,y(\sigma )) \,d\sigma , ~t \in \Lambda .\) 
